Theorem 2.5 ([1]}, Schauder)  
Let \(\mathcal {X}\)  be a Banach space, let \(\mathcal {U}\)  be a nonempty convex bounded closed subset of \(\mathcal {X}\)  and let \(\mathcal {A}:\mathcal {U}\rightarrow \mathcal {U}\)  be a completely continuous operator. Then \(\mathcal {A}\)  has at least one fixed point.

